Introduction

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Introduction

• It’s a mathematical tool for signal decomposition, like Fourier’s Transform.
• Just as the Fourier transform decomposes a signal into a series of sine and cosine functions, the wavelet transform does so using a set of functions known as wavelets.
• Wavelets are functions generated by scaling and shifting a base function known as the mother wavelet.

Introduction

Introduction

• Morlet: Popular for time-frequency analysis in EEG and ECG.
• Mexican Hat (Ricker): Often used in spike detection in neural signals.
• Haar: Useful in quick decomposition of signals and feature extraction.
• Daubechies: Frequently used in ECG signal denoising and compression.
• Symlet: Another option for signal processing and feature extraction in EEG.
• Coiflet: Useful for denoising and baseline correction in biomedical signals.

Introduction

A wavelet transformation

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A wavelet transformation

Mathematical Expressions

The continuous wavelet transform of a signal \(f(t)\) is defined as:

\[W_f(a, b) = \int_{-\infty}^{\infty} f(t) \, \psi^*\left(\frac{t - b}{a}\right) \, dt\]

where:

• \(f(t)\) is the input signal,
• \(\psi\) is the mother wavelet,
• \(a\) is the scale parameter (controls the width of the wavelet),
• \(b\) is the translation parameter (controls the position of the wavelet),
• \(\psi^*\) denotes the complex conjugate of the mother wavelet. ## Mathematical Expressions

The continuous wavelet transform of a signal \(f(t)\) is defined as:

\[W_f(a, b) = \int_{-\infty}^{\infty} f(t) \, \psi^*\left(\frac{t - b}{a}\right) \, dt\]

where:

• \(f(t)\) is the input signal,
• \(\psi\) is the mother wavelet,
• \(a\) is the scale parameter (controls the width of the wavelet),
• \(b\) is the translation parameter (controls the position of the wavelet),
• \(\psi^*\) denotes the complex conjugate of the mother wavelet.

Mathematical Expressions

The inverse continuous wavelet transform is given by:

\[f(t) = \frac{1}{C_{\psi}} \int_{0}^{\infty} \int_{-\infty}^{\infty} W_f(a, b) \, \psi\left(\frac{t - b}{a}\right) \frac{db \, da}{a^2}\]

where:

where \(C_{\psi}\) is a normalization constant, defined as:

\[C_{\psi} = \int_{0}^{\infty} \frac{|\hat{\psi}(\omega)|^2}{\omega} \, d\omega\]

and \(\hat{\psi}(\omega)\) is the Fourier transform of the wavelet \(\psi(t)\).

Mathematical Expressions

The discrete wavelet transform decomposes the signal at discrete levels of scale. For a signal \(x[n]\), the wavelet decomposition is defined as:

\[c_{j, k} = \sum_{n} x[n] \, \psi_{j, k}(n)\]

where:

• \(\psi_{j, k}(n)= \frac{1}{\sqrt{2}}\psi\left(\frac{n-2^{j}k}{2^{j}}\right)\) represents the scaled and translated versions of the mother wavelet \(\psi\)
• \(j\) is the scale index, and \(k\) is the translation index.

The decomposition typically consists of approximation and detail coefficients at each scale:

Approximation coefficients \(a_j\): capture the low-frequency components. Detail coefficients \(d_j\) capture the high-frequency components.

Mathematical Expressions

The inverse discrete wavelet transform reconstructs the original signal from its approximation and detail coefficients: \[x[n] = \sum_{j} \sum_{k} c_{j, k} \, \psi_{j, k}(n)\]

This reconstruction process involves upsampling and filtering of the approximation and detail coefficients at each scale.

Using CWT

• Purpose: The CWT is used when you need a highly detailed, continuous analysis of a signal over all possible scales and positions.
• Output: CWT gives you a “heatmap” of wavelet coefficients, showing which frequencies (or scales) are present in the signal at each point in time. This allows for a continuous representation.
• Applications: CWT is useful for analyzing signals where you want to see the evolution of frequencies over time, such as:
  • Detecting subtle changes in frequencies over time (like brainwave analysis in EEG).
  • Signals with non-repeating, transient features (like spikes in biomedical signals, e.g., ECG).
• Trade-Off: CWT is more computationally expensive because it analyzes all scales and translations continuously. It gives lots of data but is slower and requires more memory.

Using DWT

• Purpose: The DWT is used when you want a compact, efficient representation of a signal, usually for compression or feature extraction. It analyzes only specific scales (powers of two), not continuously.
• Output: DWT gives you a set of coefficients at each level (or scale), capturing information at that specific scale. It’s efficient and uses fewer data points.
• Applications: DWT is ideal when you want to reduce the size of data or focus on a smaller set of frequencies, such as:
  • Image and audio compression (like JPEG 2000 or MP3 formats).
  • Feature extraction for machine learning (e.g., identifying specific patterns).
  • De-noising signals by discarding certain scales that contain noise.
• Trade-Off: DWT is computationally cheaper but less detailed than CWT. It doesn’t give a continuous heatmap but rather a discrete set of scales.